(the first line is the result from question 5, so here I'm only asking for the answers of question 6

OP (S,, t) = -D(-d, (t)) and OP (S, , t) = KQ(-d, (t)) as, ad (t, T) 6. The Black's formula for put options is at time t is P. (t) = P(t,T) [KD(-d, (t)) -F,Q(-d, (t))]= P(t,T) g (F,), Prove that 6.1. (4) dP, (t) = 8(F,)dP(t,T) -D(-d, (t))P(t + dt,T)dF, 6.2. (4) dP, (t) = KQ(-d, (t))dP(t,T)-@(-d, (t))dS, for S, = P(t,T) F (Note: You may utilize the results of Problem 5.)